An equiconcave lens has power P. Find power of plano concave lens when given lens is cut in such a way that two plano concave lens are formed.

Options

A. `P`
B. `P/2`
C. `2P`
D. `P/4`

Answer

To solve the problem, we need to determine the power of a plano-concave lens formed by cutting an equiconcave lens with power \( P \) into two equal parts. Let's break this down step-by-step.

### Step 1: Understanding the Equiconcave Lens
An equiconcave lens is a lens that has two identical concave surfaces. The power \( P \) of the lens is related to its focal length \( F \) by the formula:
\[ P = \frac{1}{F} \]

### Step 2: Lens Maker's Formula for Equiconcave Lens
For an equiconcave lens, the lens maker's formula can be expressed as:
\[ P = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( n \) is the refractive index of the lens material, \( R_1 \) is the radius of curvature of the first surface (negative for concave), and \( R_2 \) is the radius of curvature of the second surface (also negative for concave).

For an equiconcave lens with radius of curvature \( r \):
\[ P = (n - 1) \left( \frac{-1}{r} - \frac{-1}{r} \right) = (n - 1) \left( \frac{-2}{r} \right) \]
Thus,
\[ P = \frac{-2(n - 1)}{r} \]

### Step 3: Cutting the Lens
When the equiconcave lens is cut into two plano-concave lenses, each plano-concave lens will have one flat surface and one concave surface. The power of a plano-concave lens can be calculated using the modified lens maker's formula:
\[ P' = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
In this case:
- \( R_1 = -r \) (the concave surface)
- \( R_2 = \infty \) (the flat surface)

Thus, the power of one plano-concave lens becomes:
\[ P' = (n - 1) \left( \frac{1}{-r} - 0 \right) = \frac{-(n - 1)}{r} \]

### Step 4: Relating the Powers
From the earlier equation for the equiconcave lens, we have:
\[ P = \frac{-2(n - 1)}{r} \]
Now, we can express \( P' \) in terms of \( P \):
\[ P' = \frac{-(n - 1)}{r} = \frac{1}{2} \left( \frac{-2(n - 1)}{r} \right) = \frac{P}{2} \]

### Final Result
Thus, the power of each plano-concave lens formed from the equiconcave lens is:
\[ P' = \frac{P}{2} \]

### Conclusion
The power of each plano-concave lens formed by cutting the equiconcave lens is \( \frac{P}{2} \).

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